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# Evaluation of Advanced Clustering Methods

**Date:** March 2025
**Collaborators:** [Álvaro Martínez Gamo](https://alvariitosw.github.io/portfolio_personal/)
**Technologies:** Python, Scikit-Learn, Pandas, SciPy

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## Project Overview

A systematic evaluation of four clustering algorithms (K-Means, Fuzzy C-Means (FCM), Gaussian Mixture Models (GMM), and Spectral Clustering) across synthetic datasets with varying geometries and noise levels, plus a real-world German Credit dataset. The study implements custom versions of FCM and GMM from scratch, compares them against scikit-learn baselines, and applies a battery of internal and external validation metrics to assess cluster quality and determine optimal cluster counts.

## Motivation

Standard K-Means assumes convex, isotropic clusters and fails on non-linear geometries like concentric circles or interleaved moons. Fuzzy clustering and probabilistic mixtures offer softer assignments, while spectral methods leverage graph Laplacians to detect arbitrarily shaped clusters. The goal was to understand where each method succeeds, where it breaks, and how to validate results when ground truth is available versus when it is not.

## Datasets

Three synthetic generators were used to control cluster shape and noise:

| Dataset     | Geometry                | Challenge                            |
| ----------- | ----------------------- | ------------------------------------ |
| **Blobs**   | Convex, spherical       | Baseline; all methods should succeed |
| **Circles** | Concentric, non-convex  | K-Means/FCM/GMM expected to fail     |
| **Moons**   | Interleaved, non-convex | Spectral clustering should excel     |

Noise levels of 0.05, 0.10, and 0.20 were injected to measure robustness. The German Credit dataset (1,000 samples, 20 features, binary class) provided a high-dimensional real-world test case after PCA dimensionality reduction.

![Advanced Methods Comparison.](../../../assets/projects/clustering/advanced-method-evaluation/datasets.webp)

## Algorithms

### Fuzzy C-Means

A custom implementation of FCM with configurable fuzzification parameter $m = 2$. Memberships are updated via:

$$u_{ij} = \frac{1}{\sum_{k=1}^{K} \left( \frac{d_{ij}}{d_{ik}} \right)^2}$$

Unlike hard K-Means, FCM allows points to belong partially to multiple clusters. On blob data, FCM converges to nearly identical results as K-Means; the fuzziness provides no advantage on well-separated convex clusters.

### Gaussian Mixture Models

Both scikit-learn's `GaussianMixture` and a custom EM implementation were evaluated. The custom version follows the standard E-step (responsibility computation) and M-step (mean/covariance updates) without regularization shortcuts. On blob data, both implementations achieve perfect Adjusted Rand Index (ARI = 1.0) with the correct number of components.

### Spectral Clustering

The method constructs an affinity matrix, computes the graph Laplacian, extracts the $k$ smallest eigenvectors, and applies K-Means in the spectral embedding space. This is the only method tested that reliably separates circles and moons, as it does not assume Euclidean geometry in the original feature space.

![Spectral Clustering Results.](../../../assets/projects/clustering/advanced-method-evaluation/spectral.webp)

## Quality Metrics

A suite of six metrics was applied, distinguishing internal (no labels) from external (labels available) evaluation:

| Metric                        | Type     | Measures                                      | Best Value |
| ----------------------------- | -------- | --------------------------------------------- | ---------- |
| **Silhouette Score**          | Internal | Cohesion vs. separation                       | 1.0        |
| **Davies-Bouldin Index**      | Internal | Cluster compactness / centroid distance       | 0.0        |
| **Adjusted Rand Index (ARI)** | External | Agreement with ground truth                   | 1.0        |
| **Homogeneity**               | External | Single-class purity per cluster               | 1.0        |
| **Completeness**              | External | Single-cluster purity per class               | 1.0        |
| **V-Measure**                 | External | Harmonic mean of homogeneity and completeness | 1.0        |

On synthetic blobs with 4 true clusters, all metrics correctly identify $k = 4$ as optimal: Silhouette peaks at 0.748, Davies-Bouldin bottoms at 0.354, and ARI/V-Measure both reach 1.0.

## Optimal Cluster Selection

Three methods for determining $k$ were compared on blob data:

| Method         | Criterion                | Robustness                                     |
| -------------- | ------------------------ | ---------------------------------------------- |
| **Elbow**      | Inertia inflection point | Subjective; ambiguous on noisy data            |
| **Silhouette** | Maximize average score   | Clear peak; works across algorithms            |
| **BIC (GMM)**  | Penalized log-likelihood | Best for Gaussian data; overfits on non-convex |

All three methods correctly identified $k = 4$ on the synthetic blobs, confirming their reliability when cluster structure is clean and convex.

## Conclusions

- **Algorithm selection must match data geometry.** K-Means, FCM, and GMM excel on convex blobs; Spectral Clustering is necessary for circles and moons.
- **Noise degrades all methods.** At noise = 0.20, even Spectral Clustering produces erratic boundaries.
- **Metrics agree when structure is clear.** On synthetic data, Silhouette, Davies-Bouldin, ARI, and V-Measure all converge on the same optimal $k$.
- **Clustering is not a substitute for supervision.** On the German Credit dataset, no unsupervised method outperformed a trivial majority-class classifier, demonstrating that clustering discovers structure in feature space, not necessarily structure that correlates with downstream labels.
- **Feature importance methods converge.** ANOVA, Random Forest `feature_importances_`, and PCA loadings all ranked the same top features, increasing confidence in the interpretation.